An algorithm for one-dimensional Generalized Porous Medium Equations: interface tracking and the hole filling problem

TL;DR

This paper introduces an explicit finite difference scheme for the one-dimensional Generalized Porous Medium Equation that effectively tracks free boundaries and captures hole filling phenomena, supported by convergence analysis and numerical experiments.

Contribution

It presents a novel explicit finite difference scheme capable of tracking free boundaries and modeling hole filling in the generalized porous medium equation.

Findings

The scheme accurately tracks moving free boundaries.

Numerical results demonstrate convergence as mesh size decreases.

The method captures hole filling phenomena effectively.

Abstract

Based on results of E. DiBenedetto and D. Hoff we propose an explicit finite difference scheme for the one dimensional Generalized Porous Medium Equation $\partial_t u=\partial_{xx}^2 \Phi(u)$. The scheme allows to track the moving free boundaries and captures the hole filling phenomenon when two free boundaries collide. We give an abstract convergence result when the mesh parameter $\Delta x\to 0 $ without any error estimates, and invesigate numerically the convergence rates.